cp's OEIS Frontend

A238217 The total number of 2's in all partitions of n into an even number of distinct parts.

%I A238217 #13 Jul 05 2025 08:52:32
%S A238217 0,0,0,1,0,1,1,1,1,1,2,2,3,3,5,5,7,8,10,11,14,16,19,22,26,30,35,41,47,
%T A238217 55,63,73,84,97,110,127,145,166,188,215,243,277,313,354,400,452,508,
%U A238217 573,644,723,811,910,1018,1139,1273,1421,1586,1768,1968,2190,2436
%N A238217 The total number of 2's in all partitions of n into an even number of distinct parts.
%C A238217 The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) - (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).
%H A238217 Andrew Howroyd, <a href="/A238217/b238217.txt">Table of n, a(n) for n = 0..1000</a>
%F A238217 a(n) = Sum_{j=1..round(n/4)} A067659(n-(2*j-1)*2) - Sum_{j=1..floor(n/4)} A067661(n-4*j).
%F A238217 G.f.: (1/2)*(x^2/(1+x^2))*(Product_{n>=1} 1 + x^n) - (1/2)*(x^2/(1-x^2))*(Product_{n>=1} 1 - x^n).
%F A238217 a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jul 05 2025
%e A238217 a(12) = 3 because the partitions in question are: 10+2, 6+3+2+1, 5+4+2+1.
%t A238217 Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&Length[#] == Length[ Union[#]]&&MemberQ[#,2]&]],{n,0,50}] (* _Harvey P. Dale_, Dec 09 2014 *)
%t A238217 nmax = 100; With[{k=2}, CoefficientList[Series[x^k/(1+x^k)/2 * Product[1 + x^j, {j, 1, nmax}] - x^k/(1-x^k)/2 * Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Jul 05 2025 *)
%o A238217 (PARI) seq(n)={my(A=O(x^(n-1))); Vec(x*(eta(x^2 + A)/(eta(x + A)*(1+x^2)) - eta(x + A)/(1-x^2))/2, -(n+1))} \\ _Andrew Howroyd_, May 01 2020
%Y A238217 Column k=2 of A238451.
%Y A238217 Cf. A067659, A067661.
%K A238217 nonn
%O A238217 0,11
%A A238217 _Mircea Merca_, Feb 20 2014
%E A238217 Terms a(51) and beyond from _Andrew Howroyd_, May 01 2020